Notation

Set notation

latex typesetting for set notation

$x \in X$

x is and element of set X

Natural Numbers $\N$

subset $\subseteq$

proper subset $\subset$

empty set $\varnothing$

History of nullset notation

Power set $\mathcal{P}(X)$

Example

\(X = {1,3}\\ \mathcal{P}(X) = \{\{\emptyset\}, \{1\}, \{3\}, \{1,3\}\}\)

The power set contains $2^n$ sets where n is the number of elements in the set

Complement

Complement is denoted by using a $\mathcal{C}$ or an apostrophe $(^{\prime})$

Intersection $\cap$

Properties

Union $\cup$

Properties

De Morgan Laws

1. $\mathcal{C}(A \cap B) = \mathcal{C}A \cup \mathcal{C}B$

2. $\mathcal{C}(A \cup B) = \mathcal{C}A \cap \mathcal{C}B$

Relative complement

Usually denoted with a backslash

Logic

latex typesetting for this notation

Negation $\neg$

Conjunction $\land$

p and q
p $\land$ q

Disjunction $\lor$

p or q
p $\lor$ q

Implication $\Rightarrow$

Equivalence $(\Leftrightarrow)$

Proof by contradiction

Quantifiers

For all $\forall$

There exists $\exists$

There exists at least one

There exists only one $\exists!$

Sets of numbers

Natural numbers $\N$

Integers $\Z$

Rational numbers $\mathbb{Q}$

They are eventually periodic

Irrational numbers

They are the rest of the numbers that make up the Real numbers

Real numbers $\mathbb{R}$

Pythagoras theorem fun

$\sqrt{2} \neq \mathbb{Q}$

Pythagoras and his clan where not happy that numbers could end up being notational